So far I've had the Etruscan verb put marked down in my database as a transitive verb, but upon review I think I will call this a slight booboo. It should be a ditransitive verb, meaning that it takes two objects, one in the accusative and one in the genitive. There's an interesting formula in the Liber Linteus that deserves further examination: Cis-um pute tul θansur. The sentence seems a little vague from what I can make out of it but it's certainly tied to the ritual being mentioned, whatever that ritual is. It would have been swell if those 19th-century archaeologists (which I affectionately call 'bastards') were kind enough to pay careful attention to the surrounding context from which they, shall we say, gingerly plucked this important 'mummy wrapping' text. Its resurfacing in Egypt of all places smells particularly fishy. Oh well, what can you do, really? We all know that the Museum, despite its noble image, has sexy nocturnal trysts with the Graverobber from time to time but we have no choice but to work around this thieving adultery.
So far, by insisting fiercely on strict translations that are thoroughly cross-correlated with all other known texts, I find myself deviating further and further from the status quo. Normally this would be inauspicious but this is Etruscan studies afterall where "linguistic voodoo" is still the norm. In the formula Cis-um pute tul θansur, it seems that 'participants' or 'mourners' (θansur) 'left' (pute) 'three things' (cis) 'to a boundary stone' (tul). The ending -um is just the phrasal conjunctive meaning 'and, and now, then' which carries a train of thought from the previous sentence. If we interpret put as a ditransitive verb similar to tur 'to give', we can gain insight into Etruscan grammar. Please follow along with me on this fun morphological journey.
Looking first at the verb tur, it has already been discovered by others that the recipient to which something is given is declined in the genitive. The object being given is declined in the accusative case. Since the accusative case is unmarked in nouns, it can easily be confused with the nominative unless augmented by an optional accusative demonstrative (either cn "this" or tn "that"). We might suspect that other ditransitives work similarly. Thus, I feel inspired now to propose that put marks the recipient in the accusative case while the object offered is declined in the genitive case. Note that this is slightly different than in tur because of differences in semantics between the two verbs. So, while tur means "to give", put must mean "to leave (something) behind (to)". In this case, because put appears to have a directive nuance built in (i.e. the notion of "leaving to" is implied already by the verb without needing to mark nouns specifically for "to"), the accusative case is sufficient to handle the recipient of this action.
Wrapping it all up now, my spidey senses are telling me that Cis-um pute tul θansur means "And then the mourners left the three (things) to the boundary stone." I just need to figure out what the three things are that they left behind and whether there are similar rituals elsewhere, which isn't a small task. The worship of boundary stones or termini was common in both Etruscan and Roman societies.
 In The Etruscan Language: An Introduction on page 218, Larissa and her father Giuliano Bonfante tell us that put- or puθ- is a type of "vessel" which they whimsically connect to random Latin and Greek words, as is the typical (pseudo)methodology of Etruscanists. Since the inscription ET AT 1.41 (a.k.a. TLE 188) attests to the word puθce with a verbal suffix -ce (perfective), clearly this root isn't even a noun and the Bonfantes had this information before them back in the 1970s when the book was first published. The Bonfante team goofs again, I'm afraid.
(Mar 10 2008) Admittedly, upon posting, another interpretation comes to mind: "And then the mourners left the boundary stone to the three." The advantage of this is that the genitive case thus remains consistently used for recipients while the accusative is used for the object affected by the action. This then begs the question "Who are 'the three' that receive these gifts?" to which I'm tempted to reply "But the Capitoline Triad, of course!". So far, however this is all idle conjecture that I haven't yet pursued in greater detail. Just brainstormin', is all.
(Mar 10 2008... upon waking up) Added footnote #1 to show the typical translation published (and republished) by people like G&L Bonfante.