As tides of Alabama fans and the Ohio State faithful filled the city at New Year’s for the Sugar Bowl, and as other bowl games seemed to fill every TV screen in town, Annene Kaye fielded some unexpected calls at Beachbum Berry’s Latitude 29, the tiki restaurant and bar she and her husband Jeff “Beachbum” Berry had opened just a month earlier.

“People want to know if we had TVs, were we showing the game, and when I said no, they’d say, ‘Great, we’ll be right over; we’re trying to avoid it all,’ ” Kaye said.

This weekend, some Saints fans will be seeking the same thing. With the NFL on the verge of crowning a new Super Bowl champion on Sunday, those who fully expected the Saints to be in that number when the season began may still carry a grudge from the dispiriting campaign we witnessed.

The days ahead promise endless Super Bowl previews and predictions, team profiles and punditry over team stats, player spats, deflate-gate and “beast quakes.” To forlorn Saints fans, or anyone just fed up with football, it might seem like the world is just rubbing it in, even when we go out to eat.

The TV has become the centerpiece of many bars and has found its way into more and more restaurants. That means it’s easier than ever to catch a game over a meal, but the flip side holds that it’s harder than ever to tune it all out.

Generally speaking, if you’re going to plunk down for fine dining, you’re safe, though plenty of high-end places have added TVs in their bars too. It’s the casual, unscripted dinner out that presents more problems, with some pizzerias, gastropubs, neighborhood cafés and more ambitious restaurants now flanked with screens.

One solution has emerged, however, in the growing niche of places that split the difference between upscale lounge, casual restaurant and bar. They generally eschew high-def for high style.

Latitude 29 is a new, but prime, example. Never mind just putting the Saints out of mind; this mosaic of thatch and bamboo, sea glass, swizzles, citrus and rum can make you forget you’re even in New Orleans. The menu can be every bit as exotic as the drinks, with a burger made from pork dumpling filling and seaweed-laced bun, and sliced rib-eye escorted by a stack of purple yam, taro and shiitakes.

The Freret Street cocktail lounge Cure had a TV for a short period — call it the experimental phase — though it was never a permanent fixture.

“We would wheel it out under pressure, if someone really wanted to watch a game,” Cure co-founder Neal Bodenheimer said. “I think we used it three times in a year and just got rid of it. We were trying to be everything to everyone, but now we just hope people come here for what we offer.”

In addition to cocktails sans screens, that now also includes a bar menu designed by Adam Biderman, chef at the nearby restaurant the Company Burger. Try the country ham platter with warm biscuits and whipped cane syrup (like a sweet, spreadable glaze) for finger food approximating an all-American charcuterie board. Similarly, Cure’s sibling spots — Cane & Table in the French Quarter and Bellocq on Lee Circle — are TV-free.

Close to the Dome in the CBD, CellarDoor was packed with Who Dats before home games this fall, and eventually, the staff began deploy a pull-down screen and projector on game days to provide a football feed. But ordinarily, there are no screens to interrupt the ambiance of its historic townhouse, where the kitchen sends out shrimp morsels bound in squid-ink dumplings, fried chicken skins and adobo-glazed chicken wings stuffed with dirty rice.

There’s Oxalis in the Bywater, where a progression of bar, patio, second back bar and second, still-more-secluded patio present small, cloistered spaces more conducive to close conversation than TV sight lines. Bouligny Tavern and Ivy on Magazine Street, the Delachaise on St. Charles Avenue and Treo on Tulane Avenue in Mid-City — all are cut from the same cloth, with atmospheric rooms and menus somewhere between upscale bar and laid-back bistro. While some of these are closed on Sunday, in the run-up to the game, during the days of escalating televised hype, they still offer Saints fans succor from the flat-screen reminders of what might have been.

Still, by Sunday, maybe even some jaundiced Saints fans will be ready to sit down, take their medicine and watch the Patriots and the Seahawks battle it out. It is football, after all. Plus, there are all those commercials.