Yes, you can still use travelers checks, but few businesses will accept them, and I would not expect any staff under the age of 40 to even know what they are. You will likely end up taking them to a bank to have them exchanged for cash. A prepaid debit card would be arguably no less secure, and more convenient.
Along with other non-electronic payments systems like gift certificates and money orders, travelers checks have declined sharply in popularity. The magnetic swipe card became ubiquitous from the 1980s onward (owing to historical factors varying from the structure of telecom billing to some peculiarities of financial regulation), and the U.S. is the last major economy to use them mainly because they are so entrenched.
Travelers checks will mainly be familiar to those who work in travel and tourism. I would not expect problems in a large hotel or at a major tourist attraction like Disneyland, or in tourist-oriented restaurants or shops near those attractions or hotels. On the other hand, while it is difficult to find reliable statistics, I would not expect any independent store or restaurant to honor them outside of tourist districts, and for national chains to accept them only with some hassle as operations manuals are dusted off and managers paged.
If you are insistent on travelers checks, your best bet is to take them to a bank and exchange them for cash, though not every bank offers this service to non-accountholders, and they may charge a transaction fee. American Express Travelers Cheques offers a AETC Locator that will map banks in an area that will exchange AmEx TCs for cash, but I could not find an equivalent service for Visa.
A better non-ATM, non-credit card option would be to obtain a prepaid debit card. Visa, American Express, and MasterCard all offer such products with a magnetic stripe. Like travelers checks, if lost or stolen, the card can be canceled and the value placed on a replacement. And like a credit or debit card, it can be used at any U.S. swipe terminal (caveat: at unmanned terminals you may be asked for a billing address postal code for verification; see Is there a ZIP code I can enter when paying-at-the-pump in the USA with a foreign credit card?).
Prepaid debit cards are not without their flaws, of course. There is typically a stiff fee for ATM withdrawals, on top of the ATM fee and the network fee, and there may be a conversion fee as well if loaded in your home currency. And it is possible to steal data from a swipe card and clone the card to use it fraudulently, something that would be next to impossible for a chip card.