Relics in the altar: do we still do them? Sure, but with important guidelines:
5. The tradition in the Roman liturgy of placing relics of martyrs or other saints beneath the altar should be preserved, if possible. But the following should be noted:
- a) Such relics should be of a size sufficient for them to be recognized as parts of human bodies. Hence excessively small relics of one or more saints must not be placed beneath the altar.
- b) The greatest care must be taken to determine whether the relics in question are authentic. It is better for an altar to be dedicated without relics than to have relics of doubtful authenticity placed beneath it.
- c) A reliquary must not be placed upon the altar or set into the table of the altar; it must be placed beneath the table of the altar, as the design of the altar permits.
5a would seem to be the dealbreaker for many places. We have two relics of St Thomas Aquinas at my parish, but they are fragments and not identifiable as any part of a body. Non-Catholics, and not a few Catholics themselves might find the “recognizable body part” to be indulgent to gruesome. Personally, I think a display set into a niche for devotional purposes is better, but the placement in an altar is traditional.
Some questions: Does your church have relics? Are some placed in the altar? If so, how has the design been accomplished? Are the others accessible for devotion? Are they verified?
Catholic Sensibility 
The Directory of Popular Piety and The Liturgy (2001) dovetails as follows:
237. The Missale Romanum reaffirms the validity “of placing the relics of the Saints under an altar that is to be dedicated, even when not those of the martyrs”(325). This usage signifies that the sacrifice of the members has its origin in the Sacrifice of the altar (326), as well as symbolising the communion with the Sacrifice of Christ of the entire Church, which is called to witness, event to the point of death, fidelity to her Lord and Spouse.
Many popular usages have been associated with this eminently liturgical cultic expression. An adequate pastoral instruction of the faithful about the use of relics will not overlook:
-ensuring the authenticity of the relics exposed for the veneration of the faithful; where doubtful relics have been exposed for the veneration of the faithful, they should be discreetly withdrawn with due pastoral prudence(327)
-preventing undue dispersal of relics into small pieces, since such practice is not consonant with due respect for the human body; the liturgical norms stipulate that relics must be “of a sufficient size as make clear that they are parts of the human body”(328)
-admonishing the faithful to resist the temptation to form collections of relics; in the past this practice has had some deplorable consequences
-preventing any possibility of fraud, trafficking(329), or superstition.
I parish I was working at Dedicated a New Altar and the Grey Nuns were more than happy to donate a relic of their founder when the pastor asked for one for the occasion.
Picture 13 at the following link show the placement of the relic in an a way that follows the requirements of the rite, but still allows it to be seen.
http://www.stmparish.com/altar.html