It is possible to dump all the data in a dataset to a pure ASCII file, but a little work is required on your part to edit the result to remove a few header lines that also come with the numbers. The AFNI command is

3dAFNIto3D dataset+orig.HEAD

which will produce file dataset.3D

This file is in ASCII text, with an XML-ish element header tag at the top and a XML-ish element end tag at the end. For example, the first and last few lines, including the first 2 voxels and the last voxel, of a dataset I just did this way:

# <AFNI_3D_dataset # ni_idcode = "MCW_DMRDPREMLMV"
# ni_type = "68*float"
# ni_dimen = "64,64,16"
# ni_delta = "-3.75,3.75,-7"
# ni_origin = "118.125,-118.125,52"
# ni_axes = "S-I,A-P,L-R"
# >
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
13 10 8 12 10 7 9 12 11 9 8 12 5 7 7 8 11 12 8 9 9 10 9 11 14 8 9 11 10 10 7 8 12 11 11 11 6 8 9 9 10 7 16 10 8 8 8 10 14 10 12 8 10 13 9 8 10 9 13 10 9 7 9 12 6 15 7 12
.... MANY MORE LINES ...
2 1 3 1 1 2 2 5 2 3 5 5 7 3 1 1 1 5 2 3 4 5 1 2 3 2 2 3 3 2 1 3 2 3 1 4 2 3 5 2 2 6 3 2 1 6 2 4 6 2 5 2 2 1 4 3 3 1 2 5 1 6 2 2 4 4 4 2
# </AFNI_3D_dataset>

Each line is the time series from 1 voxel. In this file, there are 64*64*16=65536 data lines, each with 68 values (there were 68 time points in this dataset).

So, if you want to import this into a spreadsheet or some other progam that likes big tables of ASCII numbers, you first will probably want to edit the file to remove the header and footer lines that start with '#'. This could easily be done with a Unix command, such as

grep -v '#' dataset.3D > dataset_new.3D

which will leave only the data lines left (this is the "little editing work" I mentioned at the start). If you want to transpose the file so that the columns are the time courses and the rows are the voxel, an AFNI command like

1dtranspose dataset.3D dataset.1D

should work. This program will ignore the '#' lines, so the grep operation above isn't required prior to the transposing.

There is no very simple way to use this method to extract just a subset of the voxels, via a mask or something similar. Some method for this purpose could be put together using various combinations of Unix and AFNI commands, by someone sufficiently clever and motivated.

Finally, AFNI can actually read a file in the .3D format as a dataset (as long as you don't remove the header lines!).