No, skeuomorphism, as a UI tool, is used as much today as it always has been.
What has changed are visual design trends. Though related to skeuomorphism, it isn't the same thing.
The term skeuomorph isn't a well defined term. I'm going to borrow the image from Trevor's deleted answer (which, BTW, I think is a very valid answer)
Where I usually find people getting confused is the mixing the concept of using skeuomorphs with the particular visual aesthetic of the UI.
Using the example above, I'd argue that:
- both screens use skeuomorphism (both UIs are replicating pushable buttons that don't actually exist on a touch screen. Both UIs are borrowing heavily from the form factor of a physical calculators analog UI.)
- one uses a flat design aesthetic, the other a realistic aesthetic.
Flat design is a visual aesthetic. It's a form of decoration.
Skeuomorphism can be that, but it's also much more.
A skeuomorph can be styled with a flat UI aesthetic, or it can be styled with a realistic aesthetic (or any other number of styles).
Another way to put it: a drop shadow on a button is an element that can be considered a skeuomorph in and of itself. But removing the drop shadow does not mean you aren't still using a skeuomorph--namely the button.
Your question is asking if you should use 'fake textures' and the like. That's an aesthetics question. Trend wise, there's less of that particular style these days. However, a misconception of 'flat design' is that they got rid of these 'realistic styles' completely. That's not true. Let's take an iOS7 modal view:
Though you don't see fake wood grain and drop shadows, it's still borrowing from real-world elements. The overlay background is emulating frosted glass, allowing portions of the background to appear with the refracted light look of real frosted glass.
So, to answer your question: Whether or not to use fake wood grain is an aesthetic decision you'll make in terms of overall visual design that may be separate from the decision as to how many skeuomorphic elements you want to borrow for your UI/interaction design. The former decision will be more related to branding and marketing. The latter decision will be more related to user experience and usability.