Here is code from my L-system processing system. I hope the syntax is self-explaining (for someone who knows L-systems).
lsystem Hilbert3D { set iterations = 3; set symbols axiom = X; interpret F as DrawForward(10); interpret + as Yaw(90); interpret - as Yaw(-90); interpret ^ as Pitch(90); interpret & as Pitch(-90); interpret > as Roll(90); interpret < as Roll(-90); rewrite X to ^ < X F ^ < X F X - F ^ > > X F X & F + > > X F X - F > X - >; }

Source: http://malsys.cz/g/Rrl8LtQx
Unfortunately I don't know source of this concrete L-system. I probably found it using google. Also it may not be original Hilbert curve since there are more ways how to fill cube with poly-line. But I will try to explain how to construct something like this.
Hilbert curve is space-filling curve, it fills cube. So rewrite step should create cube from line. There are more ways how to create cube from lines in space. One way is this:
rewrite X to ^ F + F + F & F & F + F + F ^;
Notice, that X will yield to cube but it will not change the orientation after interpreting it (it behaves like ordinary line – orientation at beginning is the same as at the end and one step ahead). To test this behavior, rewritten X and F must end in the same place with the same orientation.
Than from cube you want larger cube. This can be achieved by copying our first cube 8 times (to all 8 vertices of cube). The lines from first iteration will "connect" our new 8 cubes to one poly-line. Sou you just need place X (which yields to cube) to appropriate places (to vertices, between edges). In following rewrite rule, X are placed randomly just to illustrate the result.
rewrite X to ^ X F X + F + X F & X F & X F + X F + X F ^ X;
The tricky part is to achieve that each cube will be generated to appropriate place (in correct orientation). You need rotate turtle before interpreting X to correct position to place the cube (which will be created from X) in correct orientation. I leave the corrections up to you :)
Hint: Imagine that F is line and X is do the same change in space like F (moves forward) but it draws cube to some direction as a side effect.
EDIT: L-system of 3D extension of Hilbert curve can be also found in The Algorithmic Beauty of Plants on page 20. This awesome book about L-systems can be downloaded from algorithmicbotany.org.