Yes, I agree that you can spend months learning how to do this or just buy the thing. It's not a unique requirement.
I should think that if you care about the 1kHz sample rate then you should also care about the simultaneity of the samples. If you are looking for data at that high resolution then it becomes really significant if one channel at the end of the chain is 500us away from the first one.
The next question is how are you ensuring that the accelerometers don't give you signal frequencies higher than this? Aliasing of frequencies above the Nyquist frequency is a big problem with these kinds of measurements.
I've created a system using ADXL345 digital accelerometers to do 400Hz sampling on groups of 4. I guess that it could be ramped up to higher frequencies. My Arduino had a lot of other time-sensitive work to do at the same time, so I expect that my system could handle that frequency if it didn't have to do anything else. I just trusted the ADXL345 internal clocks to stay in sync for my measurement period by just starting them all in the same nanosecond. I would not expect the basic ADXL345s to stay acceptably in sync more than a few seconds at 1kHz.