E-3 visa is similar to the H-1B visa, but is issued only to citizens of Australia the Australia-US Free Trade Agreement (AUSFTA).
Please see for reference IRS publication 519 - http://www.irs.gov/pub/irs-pdf/p519.pdf for rules to determine if you are Nonresident Alien or Resident Alien
A person will be considered a US resident for tax purposes if you meet the substantial presence test for calendar year 2009. To meet this test, the person must be physically present in the United States on at least:
31 days during 2009, and
183 days during the 3-year period that includes 2009, 2008, and 2007, counting:
All the days you were present in 2009, and
1/2 of the days you were present in 2008, and
1/3 of the days you were present in 2007.
As you was in the US less than 183 days in 2009 and assuming you was not in the US in 2007 and 2008 - you would be considered as non-resident for 2009 tax year. Th e situation might be different for 2010 tax year.
As a non-resident - you will be taxed only on income from US sources.
The IRS treats a source of an interest income based on your residence - see IRS publication 519 - http://www.irs.gov/pub/irs-pdf/p519.pdf I put it here for your convenience
To determine if partnership distribution is taxable for you - you need to verify the type of income reported on the K-1.
The article 14 of the US-Australia tax treaty doesn't cover situations with income from a partnership.
You might find helpful IRS publication 519 - http://www.irs.gov/pub/irs-pdf/p519.pdf
page 18: If you are a member of a partnership that at any time during the tax year is engaged in a trade or business in the United States, you are considered to be engaged in a trade or business in the United States.
page 33: If you are a foreign partner in a partnership, the partnership will withhold tax on your share of effectively connected taxable income from the partnership. The partnership will give you a statement on Form 8805, Foreign Partner's Information Statement of Section 1446 Withholding Tax, showing the tax withheld.
Let me know if you need any clarification or help.