How can I counter alien grenades?

Improving your armor should probably be a priority, improved armor will prevent grenades from killing half your soldiers in one turn, but you'll still get severe injuries, and blaster bombs will still usually kill those on the center nine tiles of the blast.

Best options

There are two ways that you can prevent losing anyone to a grenade: kill (or mind control) all of the aliens close enough to throw one (or better yet, all aliens that can see you, to prevent blaster bombs as well), or use flying armor to keep your soldiers off the ground, so they can't be hit by grenades.

Mind control is probably your best option late game: if you can scout one alien, mind controlling it will often reveal several more, allowing you to quickly get the vision you need to eliminate or take control of all nearby enemies, denying them vision of your units (especially important when going up against psionic aliens or blaster bombs).

Flying, once you have it, is a good choice for your back-row snipers that tend to get clustered tightly on the first few turns, but I find flying scouts tend to be more easily spotted by aliens and often fall to plasma fire, so use it in moderation.

Early game options

The alternatives early on are not quite as flexible: you can minimize the probability of being surprised if you get vision in all directions on turn one, you can attempt to spread your soldiers out more so fewer can be caught in a blast, or you can bring out fewer soldiers at once if your first scouts don't see nearby aliens that need killing, to minimize damage if your scouting failed.

You should be able to prevent this from becoming something that happens every time, but you can't prevent it entirely. Improved armor will prevent it from killing those not at the center of the blast, but you'll still get severe injuries.

Coping with darkness

If it's a night mission, then getting proper vision to eliminate the aliens nearby is more difficult, since the aliens' vision is not reduced by low light, but yours is, making losses to grenades much more likely. Starting your soldiers near the door with a flare in their hand and having them toss it around 18 tiles away can give you full vision on the first turn, however (both soldiers and flares provide a nine-tile radius of light high enough to see aliens, not counting the center tile, so 19 tiles is as far as you should consider throwing it, but probably a few less to prevent blind spots is better).

If the terrain is flammable (e.g. jungle), I'm also a fan of giving a solider with good strength but abysmal aim an auto-cannon, and having them fire an auto shot of incendiary ammo, starting several fires spread widely across the battlefield (the worse the soldier's aim, the more widely the fires will be spread, so this is not as effective with a good soldier). Long distance snap shots are also a good option for lighting up the battlefield before scouting further.

The best way to counter early disaster after a landing is to send out your HWPs first, and then, if they don't see anything, send out the rest of your squad and spread them out.

This becomes much easier late game as you gain psionics that you can use to mind control anything you see near the beginning.

There's really no good strategy to defend against alien grenades other than to not clump up, which is harder to do early game.