Most Catholic parishes around the world, with some rare exceptions, generally tend to be places bereft of Gregorian Chant. Happily, more and more parishes seem to be interested.
The General Instruction of the Roman Missal, the constitution on the liturgy Sancrosanctam Concilium, and even the U.S. bishop’s document Liturgical Music Today call for the faithful to learn to sing at least some of the chants. I think that while it may be impossible for most congregations to sing the chants of the Graduale Romanum, the book of chants for the Mass, it is possible for most people to sing some of the other chants of the Mass.
A good place to begin is with a slender volume known as the Liber Cantualis. This book was put together in the 1970s by the monks of the famed Abbey of St. Pierre de Solesmes to serve as a basic source for Gregorian chant repertoire. The monks of Solesmes have long been regarded as the authorities on matters of Gregorian Chant. They restored the chant in the 19th century and continue to do a great deal of research into Gregorian Chant. Most of the chants in the Liber Cantualis are quite easy to sing.
There are some additional resources. In 2003, they released a couple of recordings entitled Gregorian Melodies: Popular Chants. In two volumes, these CDs cover quite a bit of the repertoire in the Liber Cantualis. They can be obtained from Paraclete Press. Or they can be downloaded from Amazon here.
The four hymn sandwich we have become accustomed to is an aberration. Hymnody does play a role historically in Catholic worship, but not at Mass. While it is certainly permitted, if you examine the General Instruction, you will note a certain preference beginning with Gregorian Chant, moving on to polyphony, then moving on to other music.