Traditional Thai: Thai town in East Hollywood where you can also visit the Griffith Park & Observatory (1935) for great views of the Hollywood sign, and some of the great LA's Frank Llyod Wright's architecture like the Hollyhock House (1922).
Traditional Korean: Koreatown (southwest of downtown LA) - the food court at Koreatown Plaza (KTP) has some substantial Korean grub at a very reasonable prices. While there you can also visit RFK's Inspiration Park.
Grand Central Market: For some greasy spoon but delicious food. Although a tourist trap, it's good to visit at least once in a lifetime and then take the nearby historic Angels Flight Railway (1901).
Olvera Street as mentioned above to also see Siqueiros' America Tropical (1932) mural and exhibit, and the historic LA Union Station (1939) with its Art Deco, Mission/Spanish Revival architecture.
Santa Monica Pier, Ocean Drive, and the nearby 3rd Street Prominade.