No, these new attacks are not a practical concern, and not a reason to change what you're doing.
@Marsh Ray already did a good job of explaining the main reason why: 128-bit keys are already long enough that there is a tremendous safety margin here. The biclique attacks speed up exhaustive search by less than a factor of 2. We already have a safety margin of many orders of magnitude, so a 2x speedup just doesn't matter. Similarly for the AES-NI instructions; there's so much safety margin against exhaustive keysearch that a 10x or 100x speedup just doesn't matter.
In addition, there's a second reason why the new biclique attacks are not a threat in practice: they require an absolutely, totally impractical amount of chosen plaintext/ciphertext pairs. For instance, their 2126.2 attack on AES-128 requires the attacker to ask for the encryption of 288 chosen-plaintexts, all encrypted under the same AES key. No current system is ever going to give an attacker the chance to acquire anywhere near that many chosen plaintexts within our lifetime. Not gonna happen.
So these attacks are completely outside the realm of practicality. They are nothing to worry about. In practice, the way that modern systems get broken is by bypassing the crypto, not by breaking the crypto. There's no way that biclique cryptanalysis of AES is gonna be the cheapest way to break into any real-world system: no way, no how.
So, don't worry about it. While the new results on biclique cryptanalysis are a highly impressive feat of mathematics, they are not a reason to shy away from AES or to change your practices. Personally, I'd say that AES is holding up remarkably well: after a decade of intense cryptanalysis, it still looks like a solid, well-thought-out design.