Is there a way in Blender to push faces like Sketchup does?
The best I can get is doing a regular extrude and a lot of work to clean the "mess" up...
Thank you for your time!
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The Rip Fill tool comes close but it still requires a bit of cleanup to remove hidden faces.
It can be accessed either by pressing space and typing "rip fill", or you can use the keyboard shortcut ALT+v.
In the example you posted, you can now press either x, y, or z to constrain the movement to whatever axis is appropriate.
The squashed and hidden faces may cause you problems so you can remove them by first using Vertex Connect Path ,j, and now Remove Doubles should collapse them.
If you want the nGons, you can select the faces and press f for Make Face.
If the model is at some arbitrary angle, you can set the Transform Orientation to Normal and then double tap z to move the face along its normal.
If you only tap z once, the movement will still be constrained to the Global axis, the second tap activates the custom Transform Orientation.
You can't do this directly, but there is a relatively easy workaround.
You could just move all of your faces in the distance you want and then extrude all but the original face out. this way the net effect is a face that was essentially extruded in. The best part about this method is, although there is an extra step, there is absolutely no cleanup work.
The example you show is really the main scenario that produces different results in terms of Push/Pull vs Extrude. I believe mano-wii's answer is as close as you'll get with the least amount of effort using the default tools. As a former SketchUp user I sympathize with this question. At the moment there is no built-in feature that can push geometry that way and remove the degenerate geometry.
Short: Nope, nothing readily available that I'm aware of (may 2015). But I did start coding something like it, after reading the question. See below.
Long: Ofcourse! code once use forever.
A fun thing about SketchUp and Blender is that they both have a scripting API. If you really want a feature code it. I'm suprised that searches come up dry with solution, but wet with exactly the same questions ( http://blenderartists.org/forum/showthread.php?160317-sketchup-like-push-pull-followup-or-efficient-method )
Though the term 'patented behaviour' is freely thrown around I'm not sure it applies to the concept but rather the exact steps of an algorithm.
Another interesting scenario is:
Here's a work in progress script / addon, that behaves like a boolean carve, it even carves straight through if needed. Still unrefined code, testers welcome.
Starting with blender version 2.9 there is a way to Extrude Manifold (Alt+E)
This tool is very similar to Extrude Faces but enables Dissolve Orthogonal Edges by default. This causes the tool to automatically split and remove adjacent faces when extruding inwards.