K702 definitely needs an amp and a pretty strong one at that.
The line level audio output from motherboards and sound cards is strong enough to power a lot of efficient, low impedance headphones and headsets. However, there are a lot of headphones that are quite inefficient and/or have high impedance inputs, the result of which is low volume and reduced sound quality. Putting a headphone amplifier into the line allows the headphone drivers to operate at their optimal voltages allowing much higher volumes and full sound quality.
Some motherboards and most sound cards now include headphone amplifiers but that doesn't mean they are all equal. The headphone amplifiers in motherboards, Xonar DG & DGX, Sound Blaster Z are fairly good quality but not powerful enough for the more difficult to drive headphones and AKG K702 fall into this category. While their impedance is fairly low at 62ohms they are notoriously inefficient and only sound their best with a decent amplifier.
I spent a few months with the K702 and Sound Blaster Z and it just wasn't up to the task of driving them all that well. Volume was a struggle and they sounded a bit thin and weedy. Upgrading to the Asus Xonar STX cured that, high volume levels were easily reached and the sound thickened up nicely too.
For a setup used purely for music an external DAC and headphone amp is the best solution but this is strictly stereo only, so you miss out on the surround sound cues you get from a sound card. Obviously this isn't a problem for music but for gaming you definitely want those features.
My personal setup is:
Sennheiser HD 650, AKG K702 (modded) and Beyerdynamic DT990 headphones
Asus Xonar Essence STX for gaming and movies
Schiit Modi (DAC) and Schiit Vali (headphone amp) for music