Joel K's answer above covers the larger-scale system pretty accurately. But the text of the prayers themselves definitely went through some changes, and the clearest come from the רצה. Obviously, the modern text we have for רצה wouldn't have worked when the Temple was standing, considering its repeated references to the destruction of the Temple and the exile. In My People's Prayer Book, volume 2 (pp. 161–162), Rabbi Lawrence A. Hoffman reconstructs the text that would have been used in the Temple era thusly:
- First he removes any mentions of the destruction – removing the phrases וְהָשֵׁב הָעֲבוֹדָה לִדְבִיר בֵּיתֶךָ and וְתֶחֱזֶינָה עֵינֵינוּ בְּשׁוּבְךָ לְצִיּוֹן בְּרַחֲמִים, as well as the chatimah of הַמַּחֲזִיר שְׁכִינָתוֹ לְצִיּוֹן.
- Then he suggests that the addition וּתְפִלָּתָם would have postdated the destruction, reflecting the unique Rabbinic emphasis on prayer as a means of "replacing" sacrifice, and that בְאַהֲבָה would have been added at the same time since בְּרָצוֹן would have covered the acceptance of sacrifices (being oft associated with the sacrificial rites).
- Then he uses the old Eretz Yisrael rite chatimah (as recorded in the Cairo Geniza and still used among some Ashkenazim on Musaf before dukhening), שֶׁאוֹתְךָ לְבַדְּךָ בְּיִרְאָה נַעֲבוֹד, since the present tense would have been accurate in Temple times.
So he ends up with a much shorter רצה prayer – רְצֵה ה׳ אֱלֹהֵינוּ בְּעַמְּךָ יִשְׂרָאֵל, וְאִשֵּׁי יִשְׂרָאֵל תְּקַבֵּל בְּרָצוֹן, וּתְהִי לְרָצוֹן תָּמִיד עֲבוֹדַת יִשְׂרָאֵל עַמֶּךָ.בָּרוּךְ אַתָּה ה', שֶׁאוֹתְךָ לְבַדְּךָ בְּיִרְאָה נַעֲבוֹד. Which he then translates as "Find favor, Ad--ai my God, in your People Israel. And the fire-offerings of Israel, accept favorably. And may you find favor in the tamid, the sacrificial service of Israel your People. Blessed are You, Ad--ai, whom we serve with awe."
Note his suggestion that the תָּמִיד of the final clause referred to the tamid-offering, which the service surrounded. In a sense, it's the prayer of prayers in this rite. Unlike in a modern Amidah, the רצה would have almost certainly been the spiritual high point of the service – the key petitionary text.
Also, the wine-libation service is listed in Tamid as well, including the psalms of the day.
- They would give the kohen the wine.
- Two kohanim with trumpets would blast, tekiah, teruah, tekiah.
- The kohen would bend down to pour the wine.
- The cymbal would sound, flags would be waved, and the Levites would sing psalms.
- As each psalm ended, the trumpets would blast and the people would prostrate themselves.
That pretty clearly implies that more than one psalm would be sang per day. Though the psalms of the day are listed in Tamid 7:4, it seems unlikely that only those seven psalms would be sang as we do today. In the minor tractate Sofrim it lists a series of psalms that were read for each major holiday, more than one in every case. Probably different psalms would be sung each day, including but NOT limited to the seven of the week.