This will certainly be a good occasion to try to explain what it is, what it MAY achieve, the potential pros and cons of the technique, and how one has to be careful with false science and "real science done in a flawed way". This will unfortunately be only a summary of things, since a lot of different ideas and opinions exist out there!
First, a bit of history. "Oil pulling" or "oil swishing" comes from an old Ayurvedic (indian traditional medicine) text, the Chakara Samhita, where it is referred to as "Kavala Graha/Kavala Gandhoosa". Even though some like to say that the book is "3000 years old", this is not really established; it was PROBABLY written between 2000 and 2600 years ago, but it MAY have been actually written by several people AND large parts of it were probably "lost" and rewritten later, and it MAY have borrowed from a lot of other treatises, and has been described by some, in its actual form, as "an amalgam of science and superstition"...So much for clear origins. But yes, the base of the whole thing is old. In the old writings, oil pulling was claimed to cure about 30 systemic diseases, including migraine/headache, asthma, and even diabetes. In a more focused fashion, it has been a traditional Indian folk remedy used to treat tooth decay, gum diseases, halitosis or bad breath, dry mouth or throat, and to generally strengthen the teeth, the gums and the jaws. It appears that it became more publicized by a Dr. F. Karach from Russia in the 1990s.
So how is it done? A tablespoon of oil (teaspoon if you are 5 to 15 years old it seems) is used to swish between the teeth for 10 to 20 minutes (the purists usually say 20). After this long process, the oil is spit out. Even though coconut oil is the latest craze, it can be done with sesame or sunflower oil, and so say just any oil will do.
How does it work? Unclear. There is a lot of junk science out there, such as "it pulls out toxins", even coming from people with dental degrees (one is quoted as saying "Cells are covered with a lipid, or fatty, membrane, which is the cell's skin. When these cells come into contact with oil, a fat, they naturally adhere to each other". False. Fat does not adhere well to fat at all. Bacteria walls are made of phospholipids, a special hybrid molecule that is both polar and non-polar.That is why you clean hands, counters and dishes with soaps (also a polar/non-polar mix), not fat. But that is not the topic today....) Here is the way some scientists speculated about the possible mechanism of the effect of sesame oil:
"The exact mechanism of the action of oil pulling therapy is not clear. It was claimed that the swishing activates the enzymes and draws the toxins out of the blood. The bottom line is that oil pulling actually cannot pull toxins out of the blood as claimed because the oral mucosa does not act as a semi-permeable membrane to allow toxins to pass through. Sesame oil has three lignans - sesamin, sesamolin, and sesaminol - that have antioxidant properties and potentiate Vitamin E action. Sesame oil has increased polyunsaturated fatty acids and the lipid peroxidation is reduced thereby reducing free radical injury to the tissues. The mechanism by which oil pulling therapy causes plaque inhibition is not known. The viscosity of the oil probably inhibits bacterial adhesion and plaque co-aggregation. The other possible mechanism might be the saponification or the 'soap-making' process that occurs as a result of the alkali hydrolysis of fat. Sesame oil is a vegetable fat and when it is acted upon by the salivary alkali, like the bicarbonates, the soap making process is initiated. Soaps are good cleansing agents because they are effective emulsifying agents. Emulsification is the process by which insoluble fats like sesame oil can be broken down into minute droplets and dispersed in water. Emulsification greatly enhances the surface area of the oil thereby increasing its cleansing action. Sesame oil is relatively high in unsaponifiable substances. The unsaponifiable fraction, a class of substances not found in other fats (sesamin or sesamolin) can probably protect the oral cavity from infection and inflammation by its antioxidant property. These mechanisms could have been the reason for the reduction of plaque scores and colony count of the microorganisms in this study. But more studies have to be done to check and prove the antibacterial effect of the components of the sesame oil." 
Conclusion? Lots of speculation, no scientific proof. Most advocates also find a potentially useful molecule in the mix, such as the sesamin, sesaminol and sesamolin described above, or lauric acid in coconut oil. Antioxidants are currently still hot as a miracle compound, but we have seen a lot of those fads come and go in the past. Also, one should remember that because a treatment is old does not always mean it is good or beneficial (bleeding people for all type of illnesses was popular for a very long time, although it tends to be rather negative in effect...Treatments without any good or bad effect have an even better chance of hanging around for a while!)
What are the advantages of the technique? It is certainly cheap. With coconut oil, probably a nice coconut breath as a result. And it is unlikely to hurt anything, as long as proper brushing and flossing is still done. The focus it brings on the teeth and gums and oral cleaning is a plus. Basically, if it does not help, it probably does not hurt either.
The disadvantages? Time-consuming, tedious...Probably something to do while one takes a shower! Lots of people find swishing oil in mouth "gross" and it activates their gag reflex. Also, it may be a lot of wasted time doing something that may not have any real effects. It is also unlikely to have any real effect in any case for people with real issues of deeper periodontal pockets (it is hard to get in there!) and would be deleterious if they substitute oil pulling for conventional periodontal treatments.
Before leaving the subject, let's use it as an example of debunking pseudo-science, and pointing flaws in some attempts at "real science".
Pseudo-science: Some of the strongest proponents of the method give as "proof of pulling toxins out of the system" the fact that the oil becomes whitish, and increases in volume after 20 minutes. Dah....If you swish oil forcefully and incorporate air and saliva (water, proteins, immunoglobulins, glycoproteins and so on) into it and create an emulsion, OF COURSE it will tend to change color (become clearer/whiter) and increase in volume (oil in mouth is perceived by the brain as food and puts the salivary glands in overdrive!). So that part is total bunk, something that was also pointed out by the researchers quoted above. It would be nice if "toxins" just leaked out, but it is not that simple.
Attempts at real science: CNN actually covered the topic, and more interesting information can be found here, including intelligent explanations and suggestions (including a - Thank Goodness! - shorter version of the treatment) by Dr. Amala Guha, an assistant professor of immunology and medicine at the University of Connecticut, who also doubles as the founding president of The International Society for Ayurveda and Health:
All good and interesting. However, we do have to reflect on the (rare) studies that were quoted and used as support to the technique. For example, "the larger one" that showed "that oil pulling had a significant effect on plaque and gingivitis". Here is the problem. When one studies the METHOD, one sees a vast difference between the instructions given for the 45 days regimen:
"Group A", or control, was instructed to:
- Rinse the mouth with saline (??? for how long).
- Then "requested to continue with their routine oral hygiene practice".
"Group B", or oil-pulling group, was given the following instructions:
- Take one tablespoon of sesame oil in the mouth on an empty stomach in the morning
- With the mouth closed and chin up, without speed or effort, sip, suck and pull the oil through the teeth in a relaxed way, and also exercise the jaw as if chewing, for a period of 15 to 20 minutes.
- Do not gargle in the throat.
- Initially the oil is viscous but slowly it turns thin and milky white as you continue.
- Spit out as and when the mouth gets full
- Wash your mouth and teeth thoroughly.
- Drink 2 to 3 glasses of water.
With that given, if we ask you to guess which group may present a 10-20% improvement in mean plaque score and gingival index, and reduction in oral bacterial counts after 45 days, would you be surprised with Group B as the winner? Given those detailed supplemental instructions given to the B group, wouldn't one expect a slight improvement in their condition since they worked so hard on their mouth? I believe the difference would have been even less (and probably inexistent) if Group A had been as focused on their mouth care and also instructed to "wash mouth and teeth thoroughly" during that time period. So here, a case of people trying to do real science, but doing it (or reporting it?) the WRONG way!!!
The method used in the "smaller study"  ("The study group was subjected to oil pulling with sesame oil (Idhayam Oil, VVV Sons India) and the control group was given 0.12% chlorhexidine mouthwash (Rexidine, Warren India) for 1 minute every day in the morning before brushing for 10 days. The participants of both the groups were allowed to brush their teeth once daily as per their daily home oral hygiene schedule.") was even more vague...But their conclusion was that oil pulling had pretty much the same effect as the classic chlorhexidine rinse.
Ok, it is time to stop here! We could go on for a while!
I hope this has proven interesting for you!
Overall conclusion: If you want to do oil pulling, why not? It may be good for the soul (and for a coconut breath) if nothing else. But do not neglect your regular periodontal (gum) care, and keep flossing and brushing!
Christiane Ashba DMD
 Asokan S, Emmadi P, Chamundeswari R. Effect of oil pulling on plaque induced gingivitis: A randomized, controlled, triple-blind study. Indian J Dent Res 2009;20:47-51
 Saravanan D, Ramkumar S, Vineetha K. Effect of Oil Pulling with Sesame Oil on Plaque-induced Gingivitis: A Microbiological Study. J Orofac Res 2013;3(3):175-180.