Many 3D printers accept 3mm filament (or 2.85mm filament), however,
don't feel that you're limited to only using one diameter of
filament. Many people choose to use 1.75mm filament instead due to
the advantages of this smaller filament.
First of all you may want to read these two interesting articles on this subject. This article from Hackaday explains why the 3D printer industry has adopted these two different sizes of filament. This article from adafruit shows you how to modify your Ultimaker printer to accept 1.75mm filament (don't worry the modifications are easy!)
So here are the reasons why you should use 1.75mm filament rather than 3mm.
1. 1.75mm filament is far more common than 3mm due to more 3D printers requiring 1.75mm over 3mm filament. This means that there is a far greater choice of 1.75mm filaments. Indeed, if you love printing with exotic mateirals such as metal and wood filament then you'll probably be able to find these filaments in 1.75mm more easily than a 3mm alternative.
2. Due to the smaller diameter, less force is required to push the filament into the printer nozzle. This means that there is less stress on the nozzle and the extruder motor which can extend the life of your 3D printer. It also means that less energy is needed to melt the filament and therefore there is less stress on your heating element in your nozzle.
3. A smaller diameter filament can result in greater precision from your printer. For example, if a printer needed to extrude exactly 10mm of filament then a greater amount of 1.75mm filament would need to be extruded than 3mm filament. However, because the amount of filament needed correlates to the length of filament, a printer is going to more accurately extrude the 1.75mm filament than the 3mm filament. This is difficult to explain but imagine trying to weigh out 300g of sugar using sugar cubes and then imagine trying to do the same thing but with powered sugar. It is going to be far easier to get exactly 300g with the powdered sugar.
4. When 3mm filament is melted and extruded there is comparably more molten filament behind the nozzle head than with 1.75mm filament. The end result is that a 3mm filament extruder is more likely to unintentionally "ooze" filament. Therefore if you find your prints suffer from unintended blobs and strings then perhaps you need to switch to 1.75mm filament.
5. Finally, 1.75mm filament makes it easy to use all of the filament on the spool. With 3mm filament, the last few metres of filament can often be coiled tightly around the spool holder therefore making it more susceptible to breaking when unravelled.
The above are just some of the advantages of using 1.75mm filament over 3mm filament. If you are still unconvinced then perhaps you should give it a try. Most 3mm printers can print with 1.75mm filament with little or no modifications. Since 1.75mm filament is similar (and even slightly cheaper) in price to 3mm filament you have nothing to lose.