How do I use an matrix of indices to reference values in another matrix without using a loop?

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I have a matrix that contains 20 vertical temperature values. It is of size 20 X 751 X 1500. Call this matrix alltemps.
I have another matrix that contains an index into the first dimension of the temperature. It is of size 1 X 751 X 1500. Call this matrix tempindex.
I want to create another 1 X 751 X 1500 matrix that is the temperature value from alltemps at the index pointed to by tempindex for all 751 X 1500 point in alltemps. Call this new matrix savedtemp
For example at 200 X 200, the temperature values in alltemps are:
280 282 284 286 288 290 292 294 296 298 300 302 304 306 308 310 312 314 316 318
And at 200 X 200, the (index) value in tempindex is 3.
So, the value in savedtemp would be 284 at 200 X 200 (i.e. the 3rd value in the temperature dimension of alltemps at 200 X 200.)
Is there a way to make a blanket (matrix-wise) assignment for every point in in the 751 X 1500 vector like this without having to run a loop?

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