The DT770 is an over ear with ridiculously comfortable velour ear pads. The 80 ohm version can be driven by your motherboard, the 250 ohm would probably be a struggle. Also the 80 ohm version comes with a straight, 3m cable while the 250 ohm version has a 3m curly cable. The 32 ohm is £20 more expensive and comes with a 1.6m straight cable as it is aimed more at mobile use but if the cable is long enough there is no reason why it wouldn't work for you.
Your mobo audio claims it can drive up to 600 ohm headphones, which is drivel and I wish manufacturers would stop using this as a specification. How easy it is to drive a pair of headphones is governed by two factors, their input impedance and their sensitivity, so just saying it will drive 600 ohm headphones doesn't mean much. An example of this, my 300 ohm Sennheiser HD650s are actually easier to drive than my 62 ohm AKG K702s and that's solely due to the sensitivity.
I guess we can use the 600 ohm claim as a kind of ball park figure though and I would expect it to be around as powerful as the Sound Blaster Z, which also makes this claim, so it "should" be able to drive the AKG K612s adequately as they are not as current hungry as the K7xx series.
The K612 is also an open back, over the ear headphone and the velour pads are very spacious. They will have a neutral sound, so bass will sound natural rather than powerful. The DT770s on the other hand are much more focused on bass, partly due to their closed back design. The difference between open back and closed back, sonically speaking, is that with closed back the bass frequencies tend to be reflected internally, which bolsters the sounds coming direct from the speaker to the ear. The result is emphasized bass, which can sound a bit boomy. The K612 is an open back design and they radiate right out of the grills on each ear cup. The differences are that without that bass reinforcement the lower frequencies sound much more natural while the mids and highs are free to form a much better sound stage (larger and more detailed).