mosfet h-bridge schematic
First, these are power mosfets that have large gate capacitance, and TTL logic are not going to provide a fast, sharp rising edge to the mosfet inputs. If you have slow rise times on the gate, you will begin to dissipate a lot of power in the power MOSFETS, resulting in smoke and pops!
Second, the motor drive voltage is 12V, and I imagine your TTL logic voltage is 5V. So, you will not be able to drive the mosfet gates strongly enough. You should aim to drive the mosfet gates with 12V square waves unless you have a good reason not to. Check with your scope to see that the waveforms have rise times well under 100 ns.
Third, if you want to use all NMOS power MOSFETS you will need to drive the upper two mosfets differently, using a floating bootstrapped power supply. The solution would be to buy a common IC called a Half-bridge Driver or a Synchronous N-Channel MOSFET Driver. Linear Tech makes the LTC4447, but there are many others available. This will give you an idea what you are looking for. A simpler approach would be to substitute PMOS for the upper power MOSFETS. Your choice comes down to cost.
Off the top of my head, I'd recommend power mosfets with a max VDS voltage of 30V, any more than that is overkill, but the motor inductance can ring and you might see greater than 12V because of it, so you want more than 12V. You have 1.1A flowing through the motor. You don't want to have to heatsink your MOSFETS so keep the power dissipation lower than 100mW. Therefore 100mW/((1.1A)^2)=0.08Ohm rDSon, with a 12V gate drive (rDSon gets higher with lower gate drives). Use some kind of MOSFET driver chip between your control logic and the MOSFET gates. If you choose the pmos/nmos solution, use something like Intersil EL7202, if you choose the dual nmos solution, use the slightly more complex half bridge driver LTC4447
Also you will need good power supply decoupling capacitors very near the power mosfets and gate driver chips. Ceramic caps of 1uF will cut down the ringing. The datasheets explain this.