Clogged heater core, remove both hoses going to the heater core (remove from water pump not firewall) and use compressed air or garden hose to force any debris out.
Bad heater core, will be leaking all over and hard to miss, replace.
Blend air door, could be a hole in the vacuum line to the blend air door actuator, locate the BAD actuator and have someone turn the selector from AC to vent to defrost, you will see the lever moving if there are no leaks. If it doesn't move you can either locate and patch the leak (temp fix) or get a new vacuum line and replace.
Temp selector, usually controlled via a wire, locate and follow the wire from climate control center to the lever that slides, turn the selector from coldest setting to the hottest and watch for proper and free movement. If it doesn't move and the wire is intact and attached you can trying using white lithium grease or oil to lubricate it. If not attached just reattach, if broken or lubricating doesn't work then replace the wire.
Trapped air in coolant system, park going up a steep incline and let the car run for 10 minutes, also helps to "bounce" the car a few times by pressing down on the bumper a couple times. FFR getting a $4 flush kit at walmart (prestone brand) will help bleed the system of trapped air and make flushing/changing the coolant system easier. They also sell thermostats that sense and bleed trapped air from the system, depending on model they are usually only $1-$3 more than the standard thermostat.
Thermostat that's stuck open, this will eventually heat up but takes longer, quickest way to rule this problem out is if the temp gauge reads at operating temp and still no heat then this isn't your problem.
I agree with acts. Your thermostat could be sticking as well. Usually when a heater core goes bad it leaks and that's when you get the cold air and not hot. If the water and antifreeze is still at the top and not leaking your heater core is probably fine. Could be a stuck thermostat also possibly the switch inside, in the dash, that controls the temperature or temperature door is faulty. Wait. Doesn't huyndai come with 10 years 100,000 mile warranty? If it's under 100,000 miles you're in luck since its an 04 and still 2014.
The heater core needs to be flushed in both directions several times *through each heater hose.
could be a number of things. check your heater core and the hoses to the heater and make sure hot water is making it to your heater. You could have a problem with your coolant system whether it be a thermostat or even a head gasket crack which would cause coolant to get into oil resovoir or oil to get into your coolant tank. Best to he a mechanic to check it and run the tests. Maybe find a local one on Clist to work on it cheap?
The door that regulates temperature isn't moving when you adjust temperature. I'll bet the heater core is hot, but air isn't going through it.
If you have just had it flushed, then it probably has an air bubble in the system and needs "burped". This is a common and easy to fix problem that happens often when a system is drained and refilled.
Your heater core went out
Could be many things
I have 2004 Hyundai Tiburon that won't blow out hot air when the heater is on, just blows out cold. I have already added coolant and have had it flushed neither worked. Any help with what could be causing this would be much appreciated. Winter is coming and i don't want to freeze!