There are quite a few ways to play around discard decks:
Play Leyline of Sanctity to make yourself Hexproof.
Since all relevant discard effects (except combat damage effects) come from spells and abilities that require you, the player, as a target, this one card hardcounters discard decks and, incidentally, also burn and many combo decks.
Play a deck without cards whose loss would cripple your strategy/combo
Sligh comes to mind, or White Weeny, or Stompy. Anything but Combo, basically, preferably Aggro. If there are no key cards in your deck, you can't lose them to discard.
Play a deck that benefits from discard
Nowadays, there are many cards, keyword mechanics mechanics, and deck archetypes that benefit from you having few cards in hand or many in your graveyard. Cursed Scroll, Ensnaring Bridge, Crucible of Worlds, Tarmogoyf, Hellbent, Threshold, and the Reanimator archetype are examples.
Draw more cards than your opponent can make you discard
All card advantage strategies apply. While this does not directly counter targeted discard, it allows you to draw more cards and therefore more copies of your win conditions, overwhelming your opponent's capability to discard them. Of course, if you lose all of your win conditions to discard, with no way to retrieve them from the graveyard, no amount of card draw will salvage the game.
Tutor for key cards
Nearly all discard, and AFAIK all targeted discard has sorcery speed. Use instant speed tutor effects like Mystical Tutor to get you the cards you need exactly when you need them, without an opening for your opponent. That increases your flexibility and allows you to carry 1 or 2 each of a wide range of strong cards.
Use targeted recycle to get the discarded card back
Use cards like Regrowth or Noxious Revival to simply return the card you had to discard. Either you get the card back immediately, or, if you have to put it on top of your library, play the recycler at the end of the opponent's turn. If enough cards are in your graveyard, this even acts as a kind of tutor, since you may get to choose from a variety of cards. Recyclers are not wasted, either, when your opponent doesn't play discard, so they can be easily integrated into the main deck.