Order of pseudo-random numbers to form samples from multivariate distribution

I am interested in creating n samples from a d-dimensional distribution (independent uniform is fine) using a pseudo-random number generator. There are two possible strategies:

Dimension-by-dimension: Use the first n random numbers as the n samples from the first dimension, the second n random numbers as the n samples from the second dimension, etc.

Sample-by-sample: Use the first d random numbers as the first multi-dimensional sample, the next d random numbers as the second multi-dimensional sample, etc.

I understand that in principle these are identical, but seeing as pseudo-random number generators are not perfect, is one preferable to the other in very high-dimensional spaces?

For concreteness, I'm using the most common form of the Mersenne Twister. The fact that the MT is 623-distributed to 32-bit accuracy suggests to me that the sample-by-sample method may be preferable, but that is largely a guess.