We have six men, all bass or baritone

wow.

Here are the current CPDL multi-category search results for: Sacred, TTB, a cappella:

Annue Christe (Anonymous)
Ave Maria (Andrea Maria Ottavini)
Cantate Domino (Michel-Richard de Lalande)
Conditor alme siderum (Guillaume Dufay)
Congaudeant catholici (Anonymous)
Ei mein Perle, du werte Kron (Michael Praetorius)
Ezekiel saw the wheel (Traditional)
Heaven is a wonderful place (Traditional)
His foot on the Treetop (Michael Gray)
Ihesu redemptor (Anonymous)
In natali Domini (Anonymous)
Macht hoch die Tür, die Tor macht weit (Sebastian Göring)
Mater patris et filia (Antoine Brumel)
Nobis est natus (Anonymous)
Pascha nostrum (Kodex des Magister Nicolaus Leopold) (Anonymous)
Relegentur ab area (Anonymous)
Sanctus (Magister Andreas)
Sanctus (Petrus de Domarto)
Si iniquitates observaveris (Samuel Wesley)
Somerset Carol (Traditional)
Sweet hour of prayer (William Batchelder Bradbury)
Tota pulchra es (John Plummer)

and the results for: Sacred, equal voices, a cappella:

Alle psallite cum luya (Anonymous)
Canons (Wolfgang Amadeus Mozart)
Gott ist unsre Zuversicht (Georg Philipp Telemann)
In pace in idipsum (Orlando di Lasso)
Wie der Hirsch (Georg Philipp Telemann)

Lastly, you may or may not find useful material in O. Ravanello's collection (seems mostly geared for SSA or TTB; some selections may work for your ensemble if transposed down).

@CHGiffen: Thanks! Either of your Ave Maria settings looks very doable for our group.

@Aristotle Esguerra: Thanks; I never think to look at CPDL because, until now, all we do is chant and I know where to find that. I like the looks of Cantate Domino.

I think you could sing the Asola setting transposed down a whole step or two with success, making the top parts accessible for Tenors or even Tenors/Baritones. Someone with Sibelius software could even produce such a transposition with minimal effort.

@ProfKwasniewski: Thank you!

I can wholeheartedly recommend the anonymous "Alle psallite cum luya" on CPDL. It's a really fun piece to sing, with the top two voices passing little bits of melody back and forth, and the third voice basically holding down a two-note drone. Not too hard, but showy! (I don't like any of the YouTube recordings of it, though - drop me a note and I'll send you a good performance.)

Ari's CPDL list is for TBB not TTB voicings, but fortunately that is what was asked for. However, for completeness, here are the TTB sacred a cappella works at CPDL:

A solis ortus cardine a 3 (Gilles Binchois)
Adoramus te (Oreste Ravanello)
Adoramus te, Christe (Giovanni Battista Martini)
Adoramus te, Christe a 3 (Orlando di Lasso)
Adoro Te devote (Mariano Garau)
Alleluia. Laudem dicite (Hans Leo Hassler)
Amor potest - Ad amorem (Anonymous)
An Wasserflüssen Babylon (Michael Praetorius)
Angelus ad pastores ait (Claudio Monteverdi)
Anima mea liquefacta est (Guillaume Dufay)
Ave Maria (Claudio Monteverdi)
Ave Maria a 3 (Charles H. Giffen)
Ay, Santa Maria (Anonymous)
Beata viscera - Corsican chant (Anonymous)
Christe qui lux es et dies (Michael Praetorius)
Christus factus est a 3 (Giovanni Matteo Asola)
Danket dem Herren (Andreas Rauch)
Deutsche Messe (Johann Michael Haydn)
Dextera Domini (Francesco Feroci)
Dixi confitebor (Orlando di Lasso)
Eripe me a 3 (Orlando di Lasso)
Es ist das Heil (Michael Praetorius)
Festa Januaria (Anonymous)
Gabriel angelus (Mathias)
Gloria (Guillaume Dufay)
Gloria laus (Guilio Bas)
Gustate et videte (Giovanni Matteo Asola)
Hodie apparuit in Israel (Orlando di Lasso)
Hoquetus David (Guillaume de Machaut)
Huic ut - Huic ut (Anonymous)
Huron Carol - Jesous Ahatonhia (Traditional)
I am come into my garden (John Arnold)
Im Anfang war das Wort (Markus Pfandler)
In monte Oliveti (Giovanni Battista Martini)
In quacumque die (Orlando di Lasso)
Laudate Dominum (Simone Stella)
Little Lamb (George Whitefield Chadwick)
Longe mi Jesu (Claudio Monteverdi)
Magnificat quarti toni (Francesco Feroci)
Magnificat Sexti Toni a 3 (Christoph Dalitz)
Mass for Three Voices (Antonio Lotti)
Miserere - Corsican chant (Anonymous)
Missa Caroli Borromaei (C. Hegmann)
Missa I in A Minor (Antonio Lotti)
Missa pro defunctis (Giovanni Matteo Asola)
Missa tribus vocibus (Christoph Dalitz)
Nihil tibi (Francesco Soriano)
O felix anima (Giacomo Carissimi)
O pie pelicane (Hans Leo Hassler)
O salutaris hostia (Giovanni Battista Martini)
O salutaris hostia (Pasquale Pisari)
Panis angelicus (Lorenzo Donati)
Parce Domine (Menegali)
Per te sciamus, da, Patrem (Michael Praetorius)
Plange quasi virgo (Oreste Ravanello)
Pues que no puedo olvidarte (Ginés de Morata)
Regina cæli lætare (Adrian Willaert)
Regina cæli lætare (Cipriano de Rore)
Salve festa dies (Christoph Dalitz)
Sophia Nasci Fertur - O Quam Pulchra Racio - Magi Videntes Stellam (Anonymous)
Tantum ergo (Antonio Bergamo)
Tantum ergo (Giuseppe Tartini)
Tristis est anima mea (Giovanni Battista Martini)
Veni Emanuel (anonymus) (Christoph Dalitz)
Verbum caro (Orlando di Lasso)
Verbum caro factum est a 3 (Hans Leo Hassler)
Vexilla regis (Francesco Feroci)
Volsvi, Persidstii tsarie (Znamenny chant)
Voskresni Bozhe (Pyotr Ivanovich Turchaninov)

Thanks again, everyone!

WJA - I've done the "Alle psallite cum luya" in concert settings with my medieval men's trio. Judging from the text (a brief explication of the word "Alleluia"), it would probably be a great fit as an Offertory motet during a particularly joyful season - Eastertide springs to mind (and the word "Alleluia" has just been gone for all of Lent!), as well as Christmastide.

translation:
Sing “alle” and “luia”;
Sing “alle” in harmony with “luia”;
Sing “alle” with your heart
entirely devoted to God, with “luia”;
Praise the Lord!