Is this a good design for MOSFET H-Bridge?
I'm not sure why you think BJTs are significantly slower than power MOSFETs; that's certainly not an inherent characteristic. But there's nothing wrong with using FETs if that's what you prefer.
And MOSFET gates do indeed need significant amounts of current, especially if you want to switch them quickly, to charge and discharge the gate capacitance — sometimes up to a few amps! Your 10K gate resistors are going to significantly slow down your transitions. Normally, you'd use resistors of just 100Ω or so in series with the gates, for stability.
If you really want fast switching, you should use special-purpose gate-driver ICs between the PWM output of the MCU and the power MOSFETs. For example, International Rectifier has a wide range of driver chips, and there are versions that handle the details of the high-side drive for the P-channel FETs for you.
Additional:
How fast do you want the FETs to switch? Each time one switches on or off, it's going to dissipate a pulse of energy during the transition, and the shorter you can make this, the better. This pulse, multiplied by the PWM cycle frequency, is one component of the average power the FET needs to dissipate — often the dominant component. Other components include the on-state power (ID2 × RDS(ON) multiplied by the PWM duty cycle) and any energy dumped into the body diode in the off state.
One simple way to model the switching losses is to assume that the instantaneous power is roughly a triangular waveform whose peak is (VCC/2)×(ID/2) and whose base is equal to the transition time TRISE or TFALL. The area of these two triangles is the total switching energy dissipated during each full PWM cycle: (TRISE + TFALL) × VCC × ID / 8. Multiply this by the PWM cycle frequency to get the average switching-loss power.
The main thing that dominates the rise and fall times is how fast you can move the gate charge on and off the gate of the MOSFET. A typical medium-size MOSFET might have a total gate charge on the order of 50-100 nC. If you want to move that charge in, say, 1 µs, you need a gate driver capable of at least 50-100 mA. If you want it to switch twice as fast, you need twice the current.
If we plug in all the numbers for your design, we get: 12V × 3A × 2µs / 8 × 32kHz = 0.288 W (per MOSFET). If we assume RDS(ON) of 20mΩ and a duty cycle of 50%, then the I2R losses will be 3A2 × 0.02Ω × 0.5 = 90 mW (again, per MOSFET). Together, the two active FETs at any given moment are going to be dissipating about 2/3 watt of power because of the switching.
Ultimately, it's a tradeoff between how efficient you want the circuit to be and how much effort you want to put into optimizing it.
It is extremely bad practice to tie MOSFET gates together without some resistance or impedance between them. Q5 and Q3 are tied together without any separation, as well as Q2 and Q6.
If you end up driving these FETs hard (which I suspect you'll end up doing), the gates can end up ringing with each other, causing nasty high-frequency (MHz) spurious turn-on and turn-off transitions. It's best to split the needed gate resistance equally and put one resistor in series with each gate. Even a few ohms is enough. Or, you could put a ferrite bead on one of the two gates.
The pull-up resistors for the gate of the P-channel FETs is on the order of two magnitudes too large. I blew a low-frequency (< 1 kHz) H-bridge like this running with a 220 ohm pull-up; I'm now at 100 Ohms and it works OK. The problem is that this causes significant parasitic current through the pull-up when turning on the P-channel, for a loss of a full watt! Also, the pull-up resistor needs to be beefy -- I paralleled some 1/4 watts, and I run the PWM pretty low, like 300 Hz.
The reason this matters is that you need to push a lot of current into the gate for a very short while to turn the MOSFET entirely on / off. If you leave it in the "in between" state, resistance will be high enough that it heats up the device and pretty quickly lets the magic smoke out.
Also, the gate resistor for the PWM controls is way too high. It, too, needs to be on the order of 100 ohms or less to drive it fast enough. If you run PWM at kilohertz or faster, you need even more, so at that point, go for a driver IC.