Solution:

The puzzle can be solved fairly easily by trial and error, but there is a way to reduce the search space considerably.

A cube has four main diagonals. The 24 ways of orienting a cube correspond exactly to the 4! ways of permuting those four diagonals. We can split the 12 edges of a cube into four sets of three edges which are symmetrically arranged around the diagonals.

Using Y=Yellow, O=Orange, B=Brown, L=Black, we have the following sets:

Cube	Sets (in no particular order)
1 (small)	YYB	YLL	OOL	OBB
2	YYL	YBB	OOB	OLL
3	YYB	YOB	OOB	LLL
4 (large)	YLL	YOB	YBB	OOL

When the puzzle is assembled, each diagonal has a set of edges from each cube, which combined should form three rows, and so they should contain 3 edges of each colour. So we should rearrange the order of each row in the above table so that each column has three of each letter. The set of three black edges on the third cube is the most restrictive - the only sets it can be combined with have brown, so to get exactly three brown edges there must be one in each set, which leaves only one way to form this column:

Cube	First diagonal solved, rest in no particular order
1 (small)	YYB	YLL	OOL	OBB
2	OOB	YYL	YBB	OLL
3	LLL	YYB	YOB	OOB
4 (large)	YOB	YLL	YBB	OOL

There is only way to get three orange edges in the column headed by YLL, which gives:

Cube	First two diagonals solved, rest in no particular order
1 (small)	YYB	YLL	OOL	OBB
2	OOB	YBB	YYL	OLL
3	LLL	YOB	YYB	OOB
4 (large)	YOB	OOL	YLL	YBB

The last two columns are then easy:

Cube	Solved
1 (small)	YYB	YLL	OOL	OBB
2	OOB	YBB	OLL	YYL
3	LLL	YOB	YYB	OOB
4 (large)	YOB	OOL	YBB	YLL

This corresponds to the following solution: