The Teubner stylesheet is an extension to the Babel package. It can be obtained from CTAN and should be installed as per the instructions in its readme.txt. I include an RPM for use with Fedora Core 4 below.

The teubner.sty is used by placing the line:

\usepackage{teubner}
in the document preamble.

So, after adding in this the document preamble (which is editable in LyX1.4 using Document->Layout-Preamble or in LyX 1.5 with Document > Settings > LaTeX Preamble) should look like:

\usepackage[greek, english]{babel}
\usepackage{teubner}
\languageattribute{greek}{polutoniko}

It is also necessary to prepend the line

\selectlanguage{english}
before everything else in the LyX window. Just to be clear: that means before the title. Make sure that the environment is "Standard": if you add this line retrospectively LyX will set it to "Title" and it won't work.

The file teubner-doc.pdf which comes with the package contains full information on how to obtain all the diacritic marks with the greek text. It is for use with Claudio Beccari's cbgreek fonts which are installed as part of the standard Fedora Core 4 distribution.

The documentation is readable at:

xpdf /usr/share/texmf/doc/latex/teubner/teubner-doc.pdf
and more interestingly (because it has slightly different information)
less /usr/share/texmf/doc/latex/teubner/teubner.dtx

There are two different ways of obtaining polytonic Greek characters.

Method 1: \foreignlanguage
In the "\selectlanguage{greek} environment the characters are obtained using the table below (also contained in the teubner-doc documentation). An example usage of this in your document (after you've taken care of including the package in the preamble) is

This is my simple tex doc. This english passage will be followed with a Greek passage.

\selectlanguage{greek}{
H>'a|are ''is some weirdly <>acc'entut'os.
}
And now we have switched back into an english environment. Note that we needed to open and close a pair of chain-braces to wrap the text which we wanted to be interpreted in the greek language environment. This is by far the easiest way for a longish thesis written mostly in English but with passages of ancient Greek interspersed throughout.

This method is supposed to do a better job of kerning (adjusting the space between letters for improved readability and aesthetics) than the next method. This one is used in the "\textgreek" environment and is quicker and easier and does a more than adequate job most of the time. The table below shows the keycodes which are also found in teubner-doc.pdf in Table 3.

Method 2: \textgreek
This method just uses a single ERT and allows the use of a set of accented vowel macros which don't work in the foreignlanguage environment of Method 1. The use of diacritical marks from Method 1 (see page 6 of teubner-doc.pdf) does work within the \textgreek environment however.

\textgreek{type accented vowel macros here.}
. The list of accented vowel macros can be found on page 9 of teubner-doc.pdf. Using them in the foreignlanguage method described above does not work.