I would say yes, the bishops and knights would probably be slightly stronger than the rooks.
Let's start out with a look at how many squares the pieces cover on an empty board:
Traditional chess:
- 64 squares on board
- Rook: 14 squares
- Bishop: 7-13 squares
- Knight: 2-8 squares
- King: 3-8 squares
Variant:
- 512 squares on board
- Rook: 21 squares
- Bishop: 21-39 squares
- Knight: 6-24 squares
- King: 6-26 squares
If we were to simply multiply based on square control, if knight-bishop-rook used to be 3 - 3.25 - 5, it would now be 9 - 9.75 - 7.5. If we were to then divide by 3 to take into account the increased power of the king, it would be 3 - 3.25 - 2.5. So by this metric the rook would indeed be weaker.
The knight's "blind spots" would be reduced. It would still, however, take 3 turns to get to an adjacent square. A knight in the corner could reach any square in 8 moves instead of 6 due to the bigger board; the loss of that blind spot might be offset by the increased number of squares not close to the knight. And a knight would still be unable to gain or lose a tempo.
You don't mention the pawn rules; they would probably have an impact on how easily a knight outpost could be created. Once created, it seems like it would be more difficult to keep an enemy knight out. A well-placed knight with 24 possible squares it could go to would seem fairly dangerous; a knight on the 4th level could threaten at least some squares in the 2nd through 6th levels directly, and could threaten forks in some places in all the levels.
The bishop would be much more difficult to trap or restrict. A bishop isn't very "bad" if it can just go over or under the pawns, after all. Visualizing their new movement might take some getting used to. They would still be restricted to one color, of course, which does limit their power just like it does in regular chess.
A bishop can't really dominate a knight on the side of the board like it can in 2d chess. The bishop can prevent the knight from going in a particular direction for one turn, but cannot maintain this.
A pair of bishops would no longer be able to keep an enemy king on the side of the board by themselves. Bishops have always, of course, had a blind spot directly in front of them, but that could be protected by the other bishop in the pair. But now they gain additional blind spots in the squares which are diagonal in all 3 dimensions. There is a 2 bishop + king helpmate available at the corner of the board, but I'm guessing it can't be forced.
Bishops would gain the ability to break a pin while still protecting the pinned piece. If, for example, a bishop was on c1 protecting a pawn on d2 from an enemy bishop on h6, the d2 pawn would be pinned - but the c1 bishop could move to d1 on an adjacent level, which still protects the pawn but breaks the pin. Rooks would not gain such an ability; a rook on d1 with a pawn on d2 and an enemy rook on d8 does not have a square it can get to in one move where it could break the pin while still protecting the pawn. (Kings and queens already have this ability in 2d chess, and it's impossible for that situation to come up with a knight in either game; there's no way for a knight to be behind a pin while directly protecting the pinned piece.)
Rooks would lose their property of being able to reach any square in two moves on an open board; now it would take three. And, of course, a rook can no longer give checkmate with only the king to help. Back rank checkmate threats would also seem to be reduced if the enemy king can escape via the other dimension. But they are still a long-range piece which can perform pins and forks.
